But Celtic's chances of progress were ended by a late Bayer Leverkusen rally from 2-1 down to win 3-2 in Germany.
Real Betis, Lazio, Monaco, Galatasaray and Olympiakos also qualified for the knockout stages on Thursday.
Van Bronckhorst watched on from the stands as Rangers suffered a shock 3-1 defeat to Hibernian in Sunday's Scottish League Cup semifinal.
But in his first match since formally succeeding Steven Gerrard, Rangers got the two-goal victory they needed to guarantee second place in Group A thanks to early goals in both halves from Alfredo Morelos.
Leicester will travel to Napoli in two weeks' time knowing a point will be enough to progress after a first-half blitz of Legia.
Patson Daka, James Maddison and Wilfried Ndidi all scored inside 33 minutes against the Polish champions to jump from bottom of the group before kick-off to top.
All four sides still have the chance to qualify in the final games of the group as the Foxes lead Napoli and Spartak Moscow by a point with Legia one point further back.
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